Fuzzy Techniques for Subjective Workload-Score ... - Semantic Scholar

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 38, NO. 6, DECEMBER 2008

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Fuzzy Techniques for Subjective Workload-Score Modeling Under Uncertainties Mohit Kumar, Dagmar Arndt, Steffi Kreuzfeld, Kerstin Thurow, Norbert Stoll, and Regina Stoll

Abstract—This paper deals with the development of a computer model to estimate the subjective workload score of individuals by evaluating their heart-rate (HR) signals. The identification of a model to estimate the subjective workload score of individuals under different workload situations is too ambitious a task because different individuals (due to different body conditions, emotional states, age, gender, etc.) show different physiological responses (assessed by evaluating the HR signal) under different workload situations. This is equivalent to saying that the mathematical mappings between physiological parameters and the workload score are uncertain. Our approach to deal with the uncertainties in a workload-modeling problem consists of the following steps: 1) The uncertainties arising due the individual variations in identifying a common model valid for all the individuals are filtered out using a fuzzy filter; 2) stochastic modeling of the uncertainties (provided by the fuzzy filter) use finite-mixture models and utilize this information regarding uncertainties for identifying the structure and initial parameters of a workload model; and 3) finally, the workload model parameters for an individual are identified in an online scenario using machine learning algorithms. The contribution of this paper is to propose, with a mathematical analysis, a fuzzy-based modeling technique that first filters out the uncertainties from the modeling problem, analyzes the uncertainties statistically using finite-mixture modeling, and, finally, utilizes the information about uncertainties for adapting the workload model to an individual’s physiological conditions. The approach of this paper, demonstrated with the real-world medical data of 11 subjects, provides a fuzzy-based tool useful for modeling in the presence of uncertainties. Index Terms—Finite-mixture models, fuzzy filtering, fuzzy modeling, subjective workload, uncertainties.

I. I NTRODUCTION

I

N ANY automated lab, there is a concern to address the issues such as assessing the workload, monitoring the stress

Manuscript received December 6, 2007; revised March 8, 2008. First published October 7, 2008; current version published November 20, 2008. This work was supported by the Center for Life Science Automation, Rostock, Germany. This paper was recommended by Associate Editor J. Gan. M. Kumar and D. Arndt are with the Center for Life Science Automation, 18119 Rostock, Germany. S. Kreuzfeld is with the Institute of Preventive Medicine, Faculty of Medicine, University of Rostock, 18055 Rostock, Germany. K. Thurow is with the Center for Life Science Automation, 18119 Rostock, Germany, with the Institute of Automation, University of Rostock, 18119 Rostock, Germany, and also with the Institute for Measurement and Sensor Systems e.V., 18119 Rostock, Germany. N. Stoll is with the Center for Life Science Automation, 18119 Rostock, Germany, and also with the College of Computer Science and Electrical Engineering, Institute of Automation, University of Rostock, 18119 Rostock, Germany. R. Stoll is with the Institute of Preventive Medicine, Faculty of Medicine and the College of Computer Science and Electrical Engineering, University of Rostock, 18055 Rostock, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCB.2008.927712

level of the staff, assessing the physical fitness of an individual for determining his ability to complete the job assigned to him, and so on. Such issues are of great importance in studies such as man–machine interactions, relationship between the automation level and the workload in automated laboratories, etc. The physiological response of an operator can be monitored to assess the individual’s status. A number of concerned physiological parameters can be interpreted by a human expert (or a computer-based expert system) for assessing the workload. Some of the difficulties in the interpretation of physiological data are measurement errors, missing data, nonreproducibility, and uncertainties due to the different physiological behaviors of different individuals. In such situations, the role of an expert system for an intelligent data interpretation to workload assessment becomes crucial to assist the human expert in solving decision-making problems. The physiological measures of the workload are based on the idea that bodily changes will be induced by a change in the workload level. Numerous studies investigating the physiological measures [including heart rate (HR), HR variability (HRV), and electroencephalogram (EEG)] have been reported in the literature [1]–[9]. The physiological parameters can often be recorded continuously with little interference on the operator’s work activities. Mathematically speaking, the subjective workload score y is related to the physiological parameters (x1 , x2 , . . . , xn ) through mapping y = f (x) where x = [x1 x2 . . . xn ] ∈ Rn is the input vector and the modeling aim is to identify the unknown function f . In light of recently developed computational-intelligence-based techniques, several studies propose neural/fuzzy methods for the modeling of the relationship between the physiological parameters and the workload level [10]–[12]. The neural and fuzzy models have been also used for the physiological-parameters-based modeling of emotion [13] and alertness level [14]. The motivation of neural/fuzzy methods is derived from their capabilities of learning complex nonlinear relationships. The fuzzy models offer a possibility to interpret the underlying relationship in terms of linguistic rules. Rani et al. [15] use a fuzzy inference system for interpreting the autonomic nervous system activities to estimate stress quantitatively. The fuzzy systems based on a fuzzy-set theory [16], [17] are considered suitable tools for dealing with the uncertainties. This has motivated many researchers to apply fuzzy techniques in medicine [18]–[32]. The neural/fuzzy modeling of the relationship between physiological parameters and the workload level is based on the assumption that there exists an ideal set of model parameters w∗ , such that the model output M (x; w∗ ) to physiological inputs x is an approximation of the subjective workload score y.

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However, it may not be possible for a given type and structure of the model M to identify the relationships between the physiological parameters and the workload level perfectly. The part of the input–output mappings that cannot be modeled for a given type and structure of the model is what we refer to as the uncertainty. Mathematically, we have y = M (x; w∗ ) + n

(1)

where n is an uncertainty arising from the following. 1) The nonoptimal choice of the type and structure of the model M . 2) The changes in physiological signals resulting from a change in the workload level are different among individuals due to the variations in body conditions, emotional states, age, gender, etc. 3) The subjective ratings of the perceived demands of the task (i.e., subjective workload score) are assigned differently by the subjects due to their biases. In the system-identification literature, n is termed as disturbance or noise. We refer to n as an uncertainty to emphasize that it represents the uncertainties regarding optimal choices of model, individual variations, and subjective ratings which have been mathematically formulated in terms of an additive disturbance n in (1). This paper is meant for addressing the fundamental issues regarding the uncertainty handling capability of the fuzzy models stated as follows. Problem 1: How can a fuzzy filter, without making any assumptions about uncertainties, be constructed for separating the uncertainties from the modeling problem? Once the uncertainties have been filtered out, how can the knowledge about uncertainties be utilized in understanding the individual variations? How can the fuzzy filter, taking into account the individual variations, be used to predict the behavior of an individual? A fuzzy model that addresses all of the issues stated in Problem 1 is obviously of great use in the accurate and reliable assessment of the operator’s functional state. For the construction of the fuzzy models using available data, a large number of techniques have been suggested in the literature, see, e.g., [33] and [34]. However, in the presence of uncertainties in the data (that is the case in subjective workload-score modeling), some robust methods for the identification of fuzzy models have been introduced [35]–[46]. To the best of the authors’ knowledge, a solution and mathematical analysis of Problem 1 have not been provided in the literature. Our previous research, outlined in [47], solves the first part of Problem 1, where an algorithm has been suggested for the fuzzy filtering of uncertainties from the stress modeling problem. To provide a unified solution to Problem 1, we introduce a fuzzy model consisting of the rules of following structure:

whereas yi represents the output value as per the ith individual’s physiological conditions. A straightforward approach to solve Problem 1, shown in Fig. 1, is as follows. 1) Collect, for an ith indexed subject, the training input– output data pairs i DPi = {x(k), y(k)}N k=1

the filtered output is equal to yf and the (2)

Here, yf represents the filtered output, i.e., a quantity that represents M (x; w∗ ) in (1). yf is only a function of the input values and is independent of the individual variations,

(3)

where x(k) is the physiological parameters vector and scalar y(k) is the subjective workload score. The total input–output data pairs, including the data of each of the S subjects, is

TD =

S i=1

If the input belongs to a cluster having a center c, then output value for ith individual is equal to yi .

Fig. 1. Combining fuzzy filtering, finite-mixture modeling, and machine learning algorithm for the modeling of a subjective workload score in the presence of uncertainties.

DPi = {x(k), y(k)}N k=1 ,

N=

S

Ni .

(4)

i=1

2) A fuzzy filter, as suggested in [47], is constructed using total data (TD) that would filter out any uncertainties (arising from individual variations) associated to the data pairs. For a data pair with an input x(k), the fuzzy filter is used to obtain a filtered output value, denoted as yf (k). That is, data pairs {x(k), yf (k)}N k=1 follow,


KUMAR et al.: FUZZY TECHNIQUES FOR SUBJECTIVE WORKLOAD-SCORE MODELING

3)

4)

5)

6)

7)

without an exception, a trend of input–output mappings. The uncertainty associated to a data pair is assessed as n ˆ k = y(k) − yf (k). The uncertainties {ˆ nk }N k=1 and filtered output values N {yf (k)}k=1 are assumed to have been produced by a set of random sources. We estimate the parameters of these random sources via modeling the 2-D data points {zk }N ˆ k ]T ∈ R2 using finite-mixture k=1 , zk = [yf (k) n models [48], [49], i.e., we estimate the parameters of a set of probability density functions (pdfs) such that each data point zk is modeled as having been generated by one of the probabilistic models in the set. The finite-mixture modeling leads to the clustering of the data points {zk }N k=1 via the identification of which source (i.e., probabilistic model) produced each data point. Assume that C different sources, with the known pdfs, have been identified producing the data {zk }N k=1 . Out of the ith indexed subject data DPi , C different data sets DPi1 , DPi2 , . . . , DPiC can be created such that the data set DPij is associated to the jth probabilistic model as per some defined criterion, i.e., DPij contains all those pairs of DPi ’s which are generated by the jth probabilistic model with a probability greater than a predefined value. Each of the data sets DPi1 , DPi2 , . . . , DPiC can be used to train (i.e., develop) a local model. Finally, the C different local models M1 , . . . , MC are combined, as shown in Fig. 1, to estimate the final output. If the local models M1 , . . . , MC are chosen to be fuzzy with the same structure as the filter, then the overall system behavior of Fig. 1 can be described in terms of fuzzy rules of the type described in (2). Further, it is possible to provide a stochastic interpretation to the method, i.e., that it outputs, under some assumptions, the conditional expected value of the workload score.

This paper is organized as follows. Section II defines the modeling problem and evaluates the classical neural/fuzzy modeling methods. A review of the fuzzy filtering theory is provided in Section III, followed by the proposed method in Section IV. Section V states the subjective workload-score modeling results. Finally, the concluding remarks are given.

II. P ROBLEM D EFINITION AND C LASSICAL A PPROACH A. Experiments The experiments were carried out to investigate the workload of the operators working in chemistry laboratories with different levels of automation. The subjects were asked to conduct an enzymatic inhibition assay under the following workload situations. 1) Manual mode. The assay was conducted manually by the operators. The requirement was to pipette and incubate the assay plates followed by the measurement of the result. The operators were asked to complete the task fast and accurately. They worked under normal lab conditions, as well as in a glove box. 2) Partial-automation mode. The operators worked with a robotics system. The experiment procedure was divided into manual work and the programming/monitoring of the

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robotics system such that there was a switching between the two. 3) Full-automation mode. The operators worked with the robotics system. They were involved in the system programming, experimental setup arrangements, and supervisory activity. Eleven subjects were included in this paper whose physiological responses were assessed via the measurement of their HRV. HRV is a measure of the variability in HR, i.e., variability in interbeat interval (IBI), which is defined as the time, in milliseconds, between consecutive R waves of an electrocardiogram. During the experiments, the IBI signal was recorded by a Polar S810i (Polar Electro Oy, Finland; time resolution, 1 ms) HR monitor. The data were collected for about 4 h during each of the following workload situations: 1) manual mode with open workplace; 2) manual mode with glove box; 3) partialautomation mode; and 4) full-automation mode. Each of the four situations was presented to the subjects on separate days in the listed order (i.e., manual mode with open workplace on the first day and full-automation mode on the fourth day). The subjective rating score of the workload was assessed using NASA Task Load Index (TLX) [50]. The NASA TLX is a method for providing an overall workload score based on a weighted average of ratings on six subscales (mental, physical, and temporal demands; own performance; effort; and frustration). For every workload situation that lasts for 4 h, the NASA TLX was collected at four different time points during 4 h such that each time point separates the tasks of two different types within a workload situation. B. IBI Signal Processing [47] The obtained IBI signal (RR interval series) was resampled at 1 Hz to produce a uniformly sampled signal [51]. Furthermore, trends were removed from the signal using the detrending method of Tarvainen et al. [52]. The IBI signal can be analyzed using some mathematical tools (e.g., fast Fourier transform, wavelet theories, and chaos) to assess the autonomic nervous system activities. The analysis of HRV in the frequency domain could provide various information about cardiovascular control [53], [54]. We extracted the features of IBI signal in the time–frequency domain using a continuous wavelet transform [47]. The continuous wavelet transform of a signal s(t) at scale a and position b is defined by t−b 1 s s(t)ψ Wψ (a, b) = √ dt a a where ψ(t) is the wavelet function. Following the method of Kumar et al. [47], the function ψ(t) was chosen equal to the 15th derivative of the complex Gaussian function 2

f (t) = C15 e−it e−t . C15 is such that f 15 2 = 1, where f 15 is the 15th derivative of f . The frequency associated with ψ(t/a) is equal to 1/a (see [47] for a justification). Wψs (a, b) can be interpreted as an estimate of the contribution of frequencies in a band around 1/a at time around b. The same physiological parameters, as in [47], were defined based on every 3-min continuous wavelet


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analysis of the signal in the frequency range 0.01–0.5 Hz (i.e., ai ∈ [2, 100], bj ∈ [0, 3]) 0.04 p1 = MW (1/f )df 0.01 0.15

p2 =

MW (1/f )df 0.04 0.5

p3 =

MW (1/f )df 0.15

b where MW (a) = (1/(nb + 1)) nj=0 |Wψs (a, bj )|2 , bj is the time index such that b0 = 0 and bnb = 3 min. The parameters p1 , p2 , and p3 assess the average power of the signal during 3 min in very low frequency (VLF) (0.01–0.04 Hz), low-frequency (LF) (0.04–0.15 Hz), and high-frequency (HF) (0.15–0.5 Hz) bands, respectively. These powers have been largely used in the literature by associating them to the autonomic nervous system activities. The LF is affected by both sympathetic and parasympathetic activities and the HF is found dominated by the parasympathetic activity [55]. The VLF is related to factors such as temperature, hormones, etc. [56]. The ratio of the LF-to-HF power has been associated with the sympathovagal balance [56]. C. Modeling Problem The goal is to develop a model that, for a given 3-min IBI signal, predicts the corresponding subjective workload score. In particular, we want to build a neural/fuzzy model M that is characterized by the adjustable parameters vector w such that

N N yˆ = M pN 2 , p3 , HR ; w N where yˆ = (NASA TLX)/100, HRN = HR/100, and pN 2 , p3 are the values of p2 , p3 , respectively, normalized to the total N power, i.e., pN 2 and p3 represent the percentage of total spectrum power in the LF and HF ranges, respectively. Remark 1: One can argue about the choice of the input parameters p2 (LF power), p3 (HF power), and HR. These parameters have been widely studied in the literature under different situations of workload. Many other physiological parameters, e.g., EEG, skin conductance, electrodermal activity, blood pressure, etc., can serve as the model inputs. However, our concern here is not to study which of the input parameters are the most suitable ones. We want to highlight, in this paper, the issue of uncertainties for the chosen input parameters. Thus, the parameters (p2 , p3 , HR) have been chosen as an example to illustrate our approach which is valid for any choice and number of physiological inputs. Typically, the workload score is evaluated by the subjects at the end of the task of a given type. In this case, the subjective evaluation provides the average workload level that a subject observed during the task. Corresponding to a single subjective workload score, a large volume of physiological data exists, N N considering that the parameters (pN 2 , p3 , HR ) are calculated after every 3 min and the task may last long. For example, if a task lasted 90 min, then corresponding to a workload score,

N N there exist 30 rows of values (pN 2 , p3 , HR ), with each row for a 3-min analysis. Under these circumstances, we formulate the workload-modeling problem as follows. Problem 2: For an individual under a given task situation, the physiological parameters are assumed to be random variables whose pdf is estimated. Once the pdf is known, the probable values of physiological parameters, characterized by a higher probability density, are identified. The aim is to develop a model that maps the probable values of physiological parameters generated under different task situations to the corresponding subjective workload scores. To express Problem 2 mathematically, assume that x(k) = N N T [pN is a particular outcome of a 3-D 2 (k) p3 (k) HR (k)] 3 random variable X ∈ R for an individual under a given task situation. Let p(x) represent the pdf of X. Now, the probable values of the physiological parameters may be chosen from the ones where the probability density is closer to the peak of the pdf, i.e., the chosen values are characterized by a higher probability density. One possible way to define the set of probable values is as follows:

{x(k), p (x(k)) ≥ p Pmax } , Pmax = max p(x), 0 < p ≤ 1 x

i.e., all the data points where the probability density is not smaller than the p times of the maximum possible value were included in the set of probable values. The choice p = 0.6065 may be interesting because all the values lying within the unit standard deviation from the mean are included in the probable values if p(x) is assumed as Gaussian. Given the input–output data pairs {x(k), y(k)} for developing a model, the parameters of model M (i.e., vector w) are typically identified using one half of the total input–output data, and the other half is used to validate the model. For our modeling problem, we created the training and testing sets as follows. 1) The point in the 4-D space, whose coordinates correspond to the minimum values of three inputs and a minimum output value, has been taken as the reference point. 2) The Euclidean distance of each data point from the reference point is calculated, and all the points are arranged in the ascending order of their distances from the reference point. 3) Every second point in the series of ascending-order arranged points is included in the testing set and the remaining points in the training set. This division of data into training and testing is meant for the sandwiching of testing data between the training ones in the sense of a Euclidean distance. D. Modeling Performance of Some Neural/Fuzzy Techniques The statistics of the experimental data is summarized in Table I and shown in Fig. 2. Table I shows, for each of the workload situations, the means and variances of the physiologN N ical measures (pN ˆ. Fig. 2 2 , p3 , HR ) and the workload score y shows the notched box-and-whisker plots for the data where outliers have been marked as “+.” Please note that our aim here is not to show the significant differences in the value of parameters among different workload situations but to identify the mappings between physiological parameters and workload levels.



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TABLE I SOME STATISTICS OF THE EXPERIMENTAL DATA

Fig. 2.

N N Notched box-and-whisker plots for (a) LF power pN 2 , (b) HF power p3 , (c) HR , and (d) workload score.

Modeling Problem 2 for our experimentally generated data is considered, using a neural-network and a fuzzy model. Let us first consider the training of a three-layer feedforward neural network. The first layer has 15 “tansig” (i.e., with hyperbolic tangent sigmoid transfer function) neurons, the second layer has eight “tansig” neurons, and the third layer one has a “purelin” (i.e., with linear transfer function) neuron. The network was initialized with random values of weights and biases. The network was trained using three different training algorithms, namely, “scaled conjugate gradient backpropagation” (MATLAB Neural Network Toolbox command “trainscg”), “Levenberg–Marquardt backpropagation” (MATLAB Neural Network Toolbox command “trainlm”), and “Bayesian regularization backpropagation” (MATLAB Neural Network Toolbox command “trainbr”). The training of the network stops if the number of epochs exceeds 5000. Moreover, a Sugeno-type fuzzy model was trained using a built-in training algorithm in the MATLAB Fuzzy Logic Toolbox (“anfis” command). The “anfis” algorithm combines the least-squares and backpropagation gradient descent methods to identify the parameters of the fuzzy model. The structure of

TABLE II PERFORMANCE OF SOME NEURAL/FUZZY MODELING METHODS FOR MODELING WORKLOAD SCORE

the fuzzy model was generated from the training data using subtractive clustering (MATLAB Fuzzy Logic Toolbox command “genfis2” with a range of influence of the cluster center for each input and output dimension equal to 0.5). The fuzzy model was trained up to 1000 epochs. The modeling performance is assessed by computing the coefficient of determination (R2 ) and root mean-squared error on the training and testing data. Table II shows the performance of some of the standard neural/fuzzy modeling methods. We observe from Table II that the modeling techniques (“trainscg” and “trainlm”) show good performance on the training data but poor performance on the testing data. This


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TABLE III PERFORMANCE OF BAYESIAN REGULARIZED NEURAL NETWORKS FOR MODELING THE INDIVIDUAL COMPONENTS OF THE WORKLOAD SCORE

where for x ∈ X, Ai (x) is a value in the closed interval [0, 1] that represents the degree to which x belongs to Ai (i.e., ith cluster). This mapping is called a membership function of the fuzzy set. For a given input vector x, the output of the filter is calculated by aggregating the rules as K

F (x) =

αi Ai (x)

i=1 K

i=1

indicates, in the modeling problem, the presence of uncertainties. These uncertainties resulted in the overtraining of the model and, thus, a poor generalization performance (as shown by poor performance on the testing data). The Bayesian regularized neural-network and fuzzy models (trained with an “anfis” command) were not overtrained. However, their performance on the training and testing data is below the acceptable level. These difficulties in the workload-modeling problem due to the uncertainties derive a motivation of studying Problem 1. It is interesting to study the relationship between the input parameters and the subscales of the NASA TLX. The performances of the neural networks trained with the “trainbr” method for modeling the individual components of NASA TLX are shown in Table III. It was observed from Table III that the individual components cannot be modeled by evaluating the chosen physiological parameters. The reason behind this is that N N these three parameters (pN 2 , p3 , and HR ) are not sufficient to estimate the individual components (temporal demand, own performance, and frustration) of the overall workload. This might be the reason of the poor results of the overall workload modeling in Table II. III. F UZZY F ILTERING This section reviews the mathematical theory of a clusteringbased fuzzy filter from our previous works [47], [57], [58].

If x belongs to a cluster having a center c1 , then y = α1 .. . If x belongs to a cluster having a center cK then y = αK where ci ∈ Rn is the center of the ith cluster and the values α1 , . . . , αK are real numbers. Such clustering-based fuzzy mappings have been originally introduced in [59] and applied to a workload-modeling problem in [47]. The degree by which an n-D vector x belongs to the ith cluster can be defined by a fuzzy set such as Ai . Given a universe of discourse X, a fuzzy subset Ai of X is characterized by mapping Ai : X → [0, 1]

(5)

Ai (x)

The membership function Ai (x) is chosen based on some fuzzy clustering criterion. By the method of fuzzy c means (FCM), the membership function Ai (x) must satisfy [60] K

˜ Am i (x)x

2

− ci → Minimum,

x∈X i=1

K

Ai (x) = 1

i=1

where m ˜ > 1 is the fuzzifier and · denotes the Euclidean norm. The membership function that minimizes this objective function for a given choice of cluster centers {ci }K i=1 follows as FCMi (x, ⎧c1 , . . . , cK1 ) , x ∈ X \ {cj }j=1,...,K ⎪ m−1 1 K ⎪ ˜ ⎪ x−ci 2 ⎨ x−cj 2 = j=1 ⎪ 1, x = ci ⎪ ⎪ ⎩ 0, x ∈ {cj }j=1,...,K \ {ci }.

(6)

A possibilistic approach for the c-means clustering relaxes the unit sum constraint on the membership values so that Ai (x) better reflects the typicality of x to the ith cluster [61]. Another approach, called the noise clustering method, has been introduced in [62] to deal with the noisy data. This approach considers noise as a separate cluster such that the membership of x to the noise cluster is defined as 1 − K i=1 Ai (x) and the noise prototype is always at the same distance from every point in the data set. Another possible clustering criterion, assuming a noise cluster outside each data cluster, minimizes

A. Mathematical Formulation The fuzzy filter establishes the mappings between the input values and the corresponding output via the creation of different clusters in the input space, and associate to each cluster the output value. The mappings between the input values (denoted by a vector x = [x1 x2 . . . xn ]T ∈ Rn ) and the output value (denoted by a scalar y) are defined using different fuzzy rules

.

Jc (Ai (x), c1 , . . . , cK ) =

K

[Ai (x)x−ci 2

x∈X i=1

+ {1+Ai (x) log Ai (x)−Ai (x)} δi ]

where the second term in the objective function is intended to be a noise cluster. The term {1 + Ai (x) log Ai (x) − Ai (x)} may be interpreted as the degree to which x does not belong to the ith cluster and, thus, the membership of x to the noise cluster. √ If the distance of x to the cluster center ci is greater than δi , then the minimization of JC (·) forces a small value of Ai (x) and a large value of the membership of xi to the noise cluster. Therefore, one of the strategies may be to set δi equal to the distance of nearest cluster center from ci , i.e., δi = minj cj − ci 2 . Minimizing JC (Ai (x), c1 , . . . , ck ), with respect to Ai (x), leads to the following expression for the membership function: x − ci 2 . (7) RCi (x, c1 , . . . , cK ) = exp − δi The membership functions of (6) and (7) can be combined by adopting a mixed clustering criterion [63], [64]. One way to



do this is to assume that the membership function Ai has two components A1i and A2i , such that Ai =

˜ A2i Am 1i + 2 2

where A1i and A2i minimize the following constrained objective function: K m

˜ A1i (x) + A2i (x) x − ci 2 xX i=1

+ {1 + A2i (x) log A2i (x) − A2i (x)} δi ] ,

K

A1i (x) = 1.

i=1

Now, A1i will be given by (6) and A2i by (7). Thus ˜ |FCMi (x, c1 , . . . , cK )|m Ai (x, c1 , . . . , cK ) = 2 RCi (x, c1 , . . . , cK ) . (8) + 2

For any membership function Ai (x) defined by (6), (7), or (8), if we define Gi (x, c1 , . . . , cK ) =

Ai (x, c1 , . . . , cK ) K Ai (x, c1 , . . . , cK )

(9)

i=1

then the output of the fuzzy filter follows from (5) as F (x) =

K

αi Gi (x, c1 , . . . , cK ).

i=1

Introduce the notations α = [α1 · · · αK ]T ∈ RK , θ = [cT 1 ··· T (Kn) cT ] ∈ R (i.e., θ is a real vector of dimension equal to k product of K and n), and G(x, θ) = [G1 (x, θ) · · · Gk (x, θ)]T ∈ RK so that the output of the fuzzy filter for an input x can be expressed as T

F (x) = G (x, θ)α. Thus, the fuzzy filter is characterized by two different parameters vectors, namely, α and θ.

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k 2 as j=0 |ej | . The performance of any estimation strategy will be affected by three kind of unknown disturbances: 1) the energy of uncertainties kj=0 |nj |2 ; 2) the deviation of initial guess α−1 from true parameter α∗ , assessed as α∗ − α−1 2 ; 3) the deviation of {θj∗ }kj=0 from their initial guess k ∗ 2 {θj−1 }kj=0 , assessed as j=0 θj − θj−1 (here, we follow the approach of [65], where the initial guess about ∗ ). θj∗ is taken equal to the estimate of θj−1 We are concerned with a robust identification method that is least sensitive to the disturbances. Our approach to the robust identification of fuzzy filter is based on an energy-gain bounding criterion [65] min

max k {θj∗ }j=0 ,{nj }kj=0 k

GT x(j), θ∗ α∗ − GT (x(j), θj ) αj 2 j ∗ {αj ,θj }k j=0 α ,

j=0

μ−1 α∗ 2 + μ−1 θ

k j=0

θj∗ − θj−1 2 +

k j=0

(10) |nj |2

where μ and μθ are positive constants. The identification method minimizes the maximum possible value of energy gain from the disturbances to the filtering errors. Such an identification method will guarantee that small disturbances cannot lead to large filtering errors. The maximum value of energy gain (that will be minimized) is calculated over all possible finite disturbances without making any statistical assumptions about the nature of the signals. This is how robustness toward disturbances is achieved. It follows from [65] that fuzzy filter parameters, based on an energy-gain approach, are identified by performing for j = 0, . . . , k the recursions θj = arg min Ψj (θ) θ

αj = αj−1 +

μG (x(j), θj ) y(j)−GT (x(j), θj ) αj−1 1+μ|G (x(j), θj ) |2

(11) (12)

where α−1 = 0 and y(j) − GT (x(j), θ) αj−1 2 2 + μ−1 Ψj (θ) = θ θ − θj−1 . 1 + μG (x(j), θ) 2

B. Identification of Fuzzy Filter Parameters For the development of the fuzzy filter, the parameters (α, θ) must be identified using the given input–output data set {x(j), y(j)}kj=0 . Assume that there exists some true fuzzy filter characterized by parameters (α∗ , {θj∗ }kj=0 ), such that

y(j) = GT x(j), θj∗ α∗ + nj where uncertainty nj arises due to the modeling errors and data noise. Let (αj , θj ) denote an estimate of (α∗ , θj∗ ) using data {x(i), y(i)}ji=0 based on some recursive estimation strategy. The filtering error for the jth indexed data is given as

ej = GT x(j), θj∗ α∗ − GT (x(j), θj ) αj . Any estimation strategy will be considered performing well if it results in a small energy of filtering errors, which is measured

C. Mathematical Analysis The robustness properties of recursions (11) and (12) are obviously as a result of solving the energy-gain bounding criterion (10). These recursions are the same as suggested in [44] but are achieved, however, by solving a local min–max regularized least-squares estimation problem and for a simple (not the clustering based) Sugeno-type fuzzy filter. The convergence and steady-state behavior of this estimation algorithm have been studied in [44]. Further, the stability of the method for the variable learning rates (μ, μθ ) has been shown in [47]. For the sake of completeness, some of these results are briefly presented here. For this, consider a fuzzy model that fits given input–output data {x(j), y(j)}kj=0 based on y(j) = GT (x(j), θj ) α∗ + vj


(13)

1456


where θj is given by (11); α∗ is some true parameters vector (that is to be estimated); and the disturbance signal vj accommodates nj and any mismatch between estimated vector θj and some true vector θj∗ . We are interested in the analysis of estimating α∗ using (12) in the presence of a disturbance signal vj , i.e., we take αj as an estimate of α∗ at the jth time instant and analyze the algorithms. The following error measures are defined for our analysis: 1) consequent error vector α ˜ j = α∗ − αj ; αj−1 ; 2) a priori recursion error ξGj (j − 1) = GT (x(j), θj )˜ 3) a posteriori recursion error ξGj (j) = GT (x(j), θj )˜ αj . Result 1: If there exists a continuous function V : R → R≥0 , K∞ functions τ1 , τ2 , τ3 , and a K-function σ such that τ1 (|s|) ≤ V (s) ≤ τ2 (|s|) ∀s ∈ R

V ξGj (j) −V ξGj (j −1) ≤ −τ3 ξGj (j −1) +σ (|vj |) ∀ξGj (j − 1) ∈ R, ∀vj ∈ R then the estimation algorithm from {vj } to {ξGj (j)} is stable. Proof: The result follows from [66]. For the details, see [47]. Theorem 1: The estimation algorithm (11) and (12) from disturbances {vj } to errors {ξGj (j)} is stable. Proof: Define a function V (s) = s2 . It can be proved, as in [47], that

V ξGj (j)) − V (ξGj (j − 1) 2 μ G(x(j), θj )2 ξGj (j − 1) μ G (x(j), θj )2 |vj |2 − . ≤ 1 + μ G (x(j), θj )2 1 + μ G (x(j), θj )2 Define a constant π = min j

2

μ G (x(j), θj ) , 1 + μ G (x(j), θj )2

The aforementioned expression shows the convergence property in a sense that the squared norm of the error vector α ˜ j is a nonincreasing function of the time index j. IV. M ETHODOLOGY Given the total training input–output data pairs TD = {x(k), y(k)}nk=1 defined by (4), our method to handle the uncertainties in the workload-modeling problem consists of the following steps. A. Identification of the Parameters of a Fuzzy Filter A fuzzy filter is identified based on the ideas outlined in Section III. The identification method can be implemented using a Gauss–Newton-based algorithm suggested in the Appendix. For a choice of the number of rules in the fuzzy filter (i.e., number of clusters K) and initial guess about the cluster centers θ−1 , a clustering on the input data (e.g., using finitemixture models [67] can be performed. The output of the identified fuzzy filter represents the filtered output value. If we denote the parameters of the identified fuzzy filter by (αI , θI , αI ∈ RK , θI ∈ RKn ), then the filtered output value of the kth indexed data is given as

yf (k) = GT x(k), θI αI . (14) The uncertainty associated to the kth indexed data will be assessed as n ˆ k = y(k) − yf (k).

(15)

B. Gaussian-Mixture Modeling of Filtered Data and Uncertainties 0 < π < 1.

Now, it is possible to write 2

V ξGj (j) − V ξGj (j − 1) ≤ −π ξGj (j − 1) + |vj |2 . If we define K∞ functions τ1 (s) = |s| min(1, |s|), τ2 (s) = s2 + |s|, τ3 (s) = πs2 , and a K function σ(s) = s2 , then τ1 (|s|) ≤ V (s) ≤ τ2 (|s|)

V ξGj (j) −V ξGj (j −1) ≤ −τ3 ξGj (j −1) +σ (|vj |) and, thus, by Result 1, the stability result is proved. To study the convergence properties, assume that nj = 0. For this case, it was shown somewhere in [47, Sec. 3.1] that ξG (j − 1)2 j ˜ αj 2 = ˜ αj−1 2 − G (x(j), θj )2 2 ξ (j − 1)2 1 Gj + . 1 + μ G (x(j), θj )2 G (x(j), θj )2 That is ˜ αj 2 = ˜ αj−1 2 ⎛ 2 ⎞ ξG (j − 1)2 1 j − ⎝1 − . ⎠ 1 + μ G (x(j), θj )2 G (x(j), θj )2

Assume that the vector zk = [yf (k) n ˆ k ]T represents one particular outcome of a 2-D random variable Z ∈ R2 whose pdf can be written as a mixture of the Gaussian distributions p(z) =

C

aj p(z|mj , Σj )

(16)

j=1

such that 1) the mixing probabilities a1 , . . . , aC satisfy aj ≥ 0 and C j=1 aj = 1; 2) the parameters mj ∈ R2 , Σj (a 2 × 2 positive definite matrix) characterize fully the jth Gaussian component exp − 12 (z − mj )T Σ−1 j (z − mj ) p(z|mj , Σj ) = . (17) (2π)2 |Σj | An approach to the clustering of data {zk }N k=1 is to fit finitemixture models (16) to the data, where a component distribution is used to model a specific cluster, i.e., the jth cluster (with mean mj and covariance Σj ) is mathematically represented by a Gaussian distribution p(z|mj , Σj ). “Expectation maximization” (EM) is the standard algorithm [49], [68] used to fit finitemixture models to the data. In this paper, however, we use the algorithm of [67] for estimating the parameters of the mixture (16). This algorithm is capable of automatically selecting the number of components C. The algorithm, unlike EM, is less



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function Aj (yf ) represents the degree by which yf belongs to the fuzzy set Ai . The overall output yî (i.e., workload score of ith indexed subject) for input x is estimated by taking the weighted average of the output provided by each rule C i

yˆ =

j=1

w ¯j Aj (yf )Mj (x) C

j=1

Fig. 3. Gaussian-mixture modeling of data. Data points and (ellipses) level curves for the different components.

sensitive to initialization and avoids the possibility of algorithm convergence to the boundary of the parameter space. As an illustration, Fig. 3 shows the Gaussian-mixture modeling of an example data where the drawn ellipses are the level curves of the component distributions. The data points in Fig. 3 could be clustered via associating each point to one of the components. The matrix Σj in (17) could be chosen to be a diagonal matrix random variables are (i.e., the two m1j Σ1j 0 independent). If mj = and Σj = , then m2j 0 Σ2j

ˆ | m2j , Σ2j p(z|mj , Σj ) = p yf |m1j , Σ1j p n 2 (yf −m1 ) exp − 2Σ1j j p(yf |m1j , Σ1j ) = 1 2πΣj 2 (nˆ −m2 ) exp − 2Σ2j

j p n ˆ | m2j , Σ2j = . 2 2πΣj

. w ¯j Aj (yf )

We want to define the membership function Aj (yf ) in such a way that the data points belonging to the region covered by Aj (yf ) are most likely to be generated by the jth probabilistic model p(yf |m1j , Σ1j ). This is done by simply defining Aj (yf ) as follows:

j = 1, . . . , C. (20) Aj (yf ) = p yf |m1j , Σ1j , In view of this choice of the membership functions, the natural choice of the rule weight w ¯j is the prior probability of observing a data point from the jth source, i.e., w ¯j = aj . Thus, the overall output by combining the local models is given as C j=1

yî =

aj p yf |m1j , Σ1j Mj (x) C j=1

aj p

yf |m1j , Σ1j

.

(21)

D. Development of Local Models (18)

(19)

C. Combination of Local Models for Subjective Workload Modeling For the workload modeling of the ith indexed subject (i = 1, . . . , S), the knowledge of component distributions (i.e., p(z|m1 , Σ1 ), . . . , p(z|mC , ΣC )) is utilized in the development of local models (M1 , . . . , MC ) valid in the predefined operating regions. The operating regions can be represented by fuzzy sets (A1 , . . . , AC ), and the local models can be combined using a fuzzy rule base For input x, if the filtered value yf = GT (x, θI )αI is A1 , then output = M1 (x), [w ¯1 ] .. .

For input x, if the filtered value yf = GT (x, θI )αI is AC , then output = MC (x), [w ¯C ].

Here, Mj (x) denotes the jth model output for the input x, and w ¯j ∈ [0, 1] is the weight of the rule that represents the belief in the accuracy of the jth rule. The degree of fulfillment of the jth rule is given by βj (yf ) = w ¯j Aj (yf ), where the membership

i Out of the ith indexed subject data DPi = {x(k), y(k)}N k=1 , i i i C different data sets (DP1 , DP2 , . . . , DPC ) can be created such that the data set DPij is associated to the jth probabilistic model. A local model Mj , associated to the fuzzy set Aj defined by (20), is trained with an input–output data set DPij defined as

DPij = {x(k),

y(k),

1 ≤ k ≤ Ni ,

Aj (yf (k)) ≥ } 0 ≤ 1. (22)

The data set DPij contains all those data of DPi whose filtered output values belong to the fuzzy set Aj , at least by a degree of . The models M1 , . . . , MC could be of any type, namely, neural, fuzzy, or any other. If the local models M1 , . . . , MC are chosen to be fuzzy with the same structure as the filter (described in Section III-A), then the functionality of the workload model can be described in terms of the fuzzy rules of the type described in (2) of Section I. In particular, we choose Mj (x) = GT (x, θI )wji where wji = [wji (1) wji (2) · · · wji (K)]T ∈ Rk is the vector of adjustable parameters. The linear parameters vector wji is estimated with the data set DPij using a machine learning algorithm. In this setting, the overall output (21) is given as C

yî =

j=1

aj p yf |m1j , Σ1j GT (x, θI )wji C j=1

aj p yf |m1j , Σ1j


.

(23)

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We structure αI = [αI (1) · · · αI (K)]T , θI = [(cI1 )T · · · (cIk )T ]T , and G(x, θI ) = [G1 (x, θI ) · · · Gk (x, θI )]T and then introduce the following fuzzy model: If x belongs to a cluster having centre cI1 , then filtered output is equal to αI (1) and output value for ith individual C

aj p yf |m1j , Σ1j wji (1) j=1

is equal to

C j=1

aj p yf |m1j , Σ1j

,

.. . If x belongs to a cluster having centre cIK , then filtered output is equal to αI (K) and output value for ith individual C

aj p yf |m1j , Σ1j wji (K) j=1

is equal to

C j=1

aj p

1 yf |mK j , Σj

.

This fuzzy model, with the membership functions defined by (8), can be associated to (14) and (23). This can be seen after aggregating the rules, as in Section III-A, to obtain the filtered output yf = G1 (x, θI )αI (1) + · · · + GK (x, θI )αI (K) = GT (x, θI )αI which is same as (14). Here, Gi (x, θI ), as per (9), is equal to I Ai (x, θI )/( K i=1 Ai (x, θ )). Similarly, the output value for the ith individual C I j=1

i

yˆ = G1 (x, θ )

aj p yf |m1j , Σ1j wji (1) C

j=1

C I j=1

+ GK (x, θ )

C Byf = ⎣ aj p(yf |m1j ,Σ1j ) j=1

Let us define a random variable Fx as follows: Fx = j if, for some data point x ∈ Rn , the filtered value yf = GT (x, θI )αI is generated by the probabilistic model p(yf |m1j , Σ1j ) and the uncertainty value n ˆ is generated by the probabilistic model p(ˆ n|m2j , Σ2j ). In other words, Fx = j if the data point is generated by the jth probabilistic model. Let y i be a variable that denotes the subjective workload score of the ith subject. Now, the conditional expectation of y i , conditioned on yf , is given by

C

aj p

···

C

p(Fx = j|yf )E(y i |yf , Fx = j)

(25)

j=1

where p(Fx = j|yf ) is the conditional probability mass function. By Bayes’ rule

1 yf |mK j , Σj

(24)

aC p(yf |m1C ,Σ1C )

C

aj p(yf |m1j ,Σ1j )

j=1

so that (24) can be expressed as yî = GT (x, θI )Wi Byf .

p(yf |Fx = j)p(Fx = j) p(yf ) p(yf |Fx = j)p(Fx = j) = C . p(yf |Fx = j)p(Fx = j)

p(Fx = j|yf ) =

is the same as in (23). To characterize the behavior of the ith individual, let us define ⎡ wi (1) · · · wi (1) ⎤ 1 C i (2) ⎥ w1i (2) · · · wC ⎢ ⎥ Wi = ⎢ . .. ⎣ . ⎦ . . i i w1 (K) · · · wC (K) a1 p(yf |m11 ,Σ11 )

E. Probabilistic View

aj p yf |m1j , Σ1j wji (K)

j=1

⎡

Fig. 4 shows a block diagram representation of the mathematical analysis of this section. As shown in Fig. 4, the quantity Wi Byf takes the individual physiological conditions of ith individual into account. The elements of the matrix Wi are adjusted (i.e., learned) in a such a way that the system output yî is as close to the true value of workload as possible.

E(y i |yf ) =

+ ···

aj p yf |m1j , Σ1j

Fig. 4. Block diagram representation of the method. The matrix Wi is adjusted to make yî equal to the true value of workload.

⎤T ⎦

(26)

j=1

The probability p(Fx = j) is equal to aj (prior probability of observing a data point from the jth source) and p(yf |Fx = j) = p(yf |m1j , Σ1j ). Thus, (25), using (26), can be written as

p yf |m1j , Σ1j aj E(y |yf ) = E(y |yf , Fx = j) C .

j=1 p yf |m1j , Σ1j aj i

C

i

j=1

Substituting E(y i |yf , Fx = j) = yf + m2j , we have

C

p yf |m1j , Σ1j aj i 2 E(y |yf ) = y f + mj C .

j=1 p yf |m1j , Σ1j aj j=1


(27)


Lemma 1: If the adjustable parameters of the jth local model (i.e., wji ) are chosen such that

wji (1)

i

I

I

m2j

= E y |yf = α (1), Fx = j = α (1) +

i wj (2) = E y i |yf = αI (2), Fx = j = αI (2) + m2j

N

i,j , we have {G(x(k), θI )}k=1

p {y(k)} |wji,∗ , G x(k), θI =

2πΣ2j

Ni,j

(30)

k=1

(28)

then GT (x, θI )wji = yf + m2j , considering that G1 (x, θI ) + · · · + GK (x, θI ) = 1. In this case, the overall output that combines the local models, i.e., (23), in view of (27) becomes yî = E(y i |yf ).

considering that {y(k)}, conditioned on wji,∗ and {G(x(k), θI )}, is an independent Gaussian with a mean {GT (x(k), θI )wji,∗ } and a variance Σ2j . Using Bayes’ rule, the MAP estimator of wji,∗ will maximize the a posteriori probability

p wji,∗ | {y(k)} , G x(k), θI

F. Learning Algorithms Lemma 1 suggests that if the parameters of the local fuzzy i models (i.e., w1i , . . . , wC ) are chosen based on (28), then the combination of the local models provides the conditional expected value of the subjective workload score of the ith subject. However, in a case, one may be interested in choosing the parameters wji based on some estimation criterion e.g., maximum a posteriori (MAP). Assume that there exist some true parameters of the jth local model wji,∗ , such that the input–output data of set DPij are related via

=

i,∗

p wj | G x(k), θI p {y(k)} |wji,∗ , G x(k), θI p ({y(k)} | {G (x(k), θI )})

.

(31) We assume that wji,∗ is independent of {G(x(k), θI )}, i.e., p(wji,∗ |{G(x(k), θI )}) = p(wji,∗ ). Thus, the MAP estimator satisfies

i,∗ p wj . max p {y(k)} |wji,∗ , G x(k), θI wji,∗

y(k) = GT x(k), θI wji,∗ + vk where vk is the Gaussian with mean zero and variance Σ2j . Further, assume that the elements of vector wji,∗ are independent random variables with distribution functions as

p wji,∗ (1) = p y i |yf = αI (1), Fx = j

p wji,∗ (2) = p y i |yf = αI (2), Fx = j .. .

p wji,∗ (K) = p y i |yf = αI (K), Fx = j . That is, the probability density of random vector wji,∗ is given as T i,∗ i,∗ i,∗ i,∗ −1 1 w w exp − −w (0) σ −w (0) j j j j 2 wji p wji,∗ = (2π)K |σwji | (29) where wji,∗ (0) = αI + m2j σwji = Σ2j I. Our concern is to estimate the vector wji,∗ using a N

1

⎧ ⎫ i,j 2 ⎬ ⎨ 1 N

i,∗ × exp − 2 y(k) − GT x(k), θI wj ⎩ 2Σj ⎭

.. .

wji (K) = E y i |yf = αI (K), Fx = j = αI (K) + m2j

1459

i,j . Conditioned on wji,∗ and data set DPij = {x(k), y(k)}k=1

It follows when using (29) and (30) that the MAP estimator solves ⎡ ⎤ Ni,j 2 ) )2

) i,∗ ) i,∗ i,∗ min ⎣ y(k) − GT x(k), θI wj + )wj −wj (0)) ⎦ wji,∗

k=1

which could be computed recursively (i.e., in an online manner) using the well-known recursive least-squares (RLS) algorithm. Some researchers are not interested in an average or expected performance but in the worst case performance, where performance analysis is made without making any statistical assumptions on the variables. The p-norm algorithm [69] is one of the typical examples of such learning techniques. The p-norm algorithms are an online learning tool which, for p = 2, yield the Widrow–Hoff learning rule, whereas p = 2 ln K (where K is the length of parameters vector wji ) gives rise to an algorithm which is very similar to the “exponentiated gradient.” The details of these algorithms are available in, e.g., [69] and [70]. In this paper, these details are not provided in order not to loose track of the central Problem 1. Further, the choice of the learning algorithm and a mathematical analysis of its effect on the overall system performance are some of the issues which require considerable research efforts and cannot be covered in a reasonable length in this paper. V. R ESULTS We start with a simple display of the data in Fig. 5(a)–(c). As shown in Fig. 5, the workload-score prediction based on


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Fig. 5. Plots of different physiological parameters with subjective workload score. (a) Plot between pN ˆ. (b) Plot between pN ˆ. (c) Plot between 3 and y 2 and y HRN and yˆ.

the chosen physiological parameters is a tough job, i.e., a considerable amount of uncertainties exists. This explains the poor performance of the neural/fuzzy modeling techniques in Table II. We employed a fuzzy filter, with membership functions defined by (8) for m ˜ = 2, for filtering out the uncertainties from the modeling problem. The fuzzy filter parameters were identified based on the energy-gain bounding approach. The identification method was implemented in MATLAB 6.5 using a Gauss–Newton-based algorithm proposed in the Appendix. The number of rules in the fuzzy filter (i.e., K) and initial guess about the cluster centers (θ−1 ) were chosen via performing clustering on the 3-D input training data using finite-mixture models [67]. The identification algorithm was run up to ten epochs, taking μ = μθ = 0.1. The identified fuzzy filter was used to obtain the filtered values (14) and the underlying uncertainties (15). The Gaussian-mixture modeling of the 2-D data (filtered and uncertainties values) identified nine different component distributions (chosen to have a common diagonal covariance for all components). These nine component distributions have been shown in Fig. 3. The membership functions associated to the component distributions, defined by (20), were as follows: 2 (yf −m1j ) exp − 2(0.0023) , Aj (yf ) = 2π(0.0023)

j = 1, . . . , 9

where m11 = 0.2904

m12 = 0.5436

m13 = 0.2301

m14 = 0.4901

m15 = 0.5033

m16 = 0.5451

m17 = 0.4369

m18 = 0.1079

m19 = 0.2537.

The mixing probabilities were as follows: a1 = 0.1438

a2 = 0.1373

a3 = 0.0374

a4 = 0.0553

a5 = 0.2569

a6 = 0.1924

a7 = 0.0805

a8 = 0.0565

a9 = 0.0399.

The parameter in (22) was taken is such a way that all data lying at ±3 Σ1j from the mean m1j are included in the DPij . The linear parameters of the local fuzzy models were learned based on the MAP estimation criterion (i.e., RLS algorithm). Several epochs of the RLS algorithm were run until the parameters almost converged. The maximum number of epochs was 100. The modeling performance of our method is shown in the first row of Table IV and in Fig. 6(a) and (b). A comparison between the achieved performance and Table II clearly shows the effectiveness of our approach. The method achieved a prediction accuracy of R2 = 0.7144 on the testing data in comparison with the prediction accuracy of



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TABLE IV COMPARISON OF THE PROPOSED APPROACH WITH A STANDARD FUZZY MODELING TECHNIQUE

Fig. 6.

Fig. 7.

Performance of the proposed approach on (a) training data and (b) testing data.

PDF of uncertainties and its nine different Gaussian components.

R2 = 0.4627 of a Bayesian regularization neural network in Table II. These results seem to justify our approach in handling the uncertainties associated to a modeling problem. It may be interesting to analyze the statistics of the uncertainties. Fig. 7 shows the pdf of uncertainties and its nine different Gaussian components. An alternative to our uncertainty handling approach is to develop an independent model for each subject. For the sake of

comparison, the Sugeno fuzzy models, whose structures were generated by subtractive clustering (MATLAB Fuzzy Logic Toolbox command “genfis2,” with a range of influence of the cluster center for each input and output dimension varying from 0.5 to 1), were trained (using MATLAB Fuzzy Logic Toolbox “anfis” command) for each subject. The training continued up to ten epochs. The overall modeling performance on 11 subjects is listed in Table IV. The poor performance of the anfis models (whose structures were generated for the range of influences of 0.5, 0.6, 0.7, and 0.8) was due to the fact that the individual models were overtrained (because of the nonrobust nature of the training algorithm), i.e., the mathematical mappings between physiological parameters and different workload situations are uncertain even for the same subject. The anfis models (whose structures were generated for the range of influences of 0.9–1) were not overtrained; however, their performances on the testing data were poor. The comparison of the first row of Table IV with others verifies that our approach could be an effective modeling tool in the presence of uncertainties. Remark 2: The robustness of a neural/fuzzy modeling technique against uncertainties is an issue closely related to our modeling problem. The overtraining of the models is avoided via rendering robustness to the training algorithm. Please note that our approach for the handling of uncertainties was to solve Problem 1 instead of addressing the issue of robustness or sensitivity against uncertainties.


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VI. C ONCLUDING R EMARKS This paper has outlined a fuzzy- and statistical-methodsbased technique to handle the uncertainties which arise in the development of a computer model to estimate the subjective workload score of individuals by evaluating the HR signals. Due to the individual variations, the uncertainties magnitude is very high that the classical neural/fuzzy modeling methods fail to model the relationships between the subjective workload score and the physiological parameters. Our approach that combines fuzzy filtering and stochastic modeling methods to solve Problem 1 seems to be an effective method of modeling under uncertainties. There are some issues and variants of our approach. In particular, we would like to mention the following. 1) The choice and number of physiological inputs affect the modeling performance. It may be possible to get improved results if some other inputs, chosen on the basis of some mathematical criterion, are considered as model inputs. However, this is not the concern of this paper. Our concern was to handle the uncertainties once the choice of inputs has been made. 2) Instead of a fuzzy filter, some other type of filter (e.g., a neural network) could be used to filter out the uncertainties. However, fuzzy models, by virtue of their membership functions, are considered more suitable tools in the presence of uncertainties. 3) The robustness property of the learning algorithms (used for the training of local models) will affect the performance of the proposed method. A mathematical analysis of the effect of the learning algorithm’s robustness on the modeling performance is part of our future work.

A PPENDIX G AUSS -N EWTON -B ASED A LGORITHM −1 Given N input-output data pairs {x(j), y(j)}N j=0 , to compute the parameters

θj = arg min r(θ)2 θ

where ⎡ ⎢ r(θ) =⎣

2

[y(j)−GT (x(j),θ)αj−1 ]

αj =αj−1+

1+μG(x(j),θ)2

1/2 ⎤

1/2 μ−1 (θ − θj−1 ) θ

⎥ ⎦

μG (x(j), θj ) y(j)−GT (x(j), θj ) αj−1 1+μ G (x(j), θj )2

(32)

we suggest a Gauss-Newton-based algorithm. The algorithm consists of the following steps. 1) Choose an initial guess about cluster centers θ−1 , number of maximum epochs Emax , α−1 = 0, epoch count EC = 0, and data index j = 0.

2) If EC < Emax a) If j ≤ (N − 1) ⎡ * i) Define r(θ) = ⎣

+1/2 ⎤

[y(j)−GT (x(j),θ)αj−1 ]2 1+μG(x(j),θ)2 −1 1/2 μθ (θ − θj−1 )

⎦ and

let s∗ (θ) be the unique solution of following linear least-squares problem: * + 2 s∗ (θ) = arg min r(θ) + r (θ)s s

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Mohit Kumar received the B.Tech. degree in electrical engineering from the National Institute of Technology, Hamirpur, India, in 1999, the M.Tech. degree in control engineering from the Indian Institute of Technology, Delhi, in 2001, and the Ph.D. degree (summa cum laude) in electrical engineering from the University of Rostock, Rostock, Germany, in 2004. He was a Research Scientist with the Institute of Occupational and Social Medicine, Rostock, from 2001 to 2004. Currently, he is the Head of the research group “Life Science Automation—Technologies,” Center for Life Science Automation, Rostock. His research interests include modeling of complex and uncertain processes with applications to life science. He took an initiative in intelligent fuzzy computing to build a mathematical bridge between artificial intelligence and real-world applications (www.fuzzymodeling.com).

Dagmar Arndt received the Dr.Med. degree in medicine from the Semmelweis University, Budapest, Hungary, and the University of Rostock, Rostock, Germany. For a year, she was as an Assistant Doctor of the clinic at the University of Rostock. Since 2005, she has been with the Center for Life Science Automation, Rostock. Her research interests include occupational physiology and stress monitoring.

Steffi Kreuzfeld received the Dip.Med. degree and the Dr.Med. degree in medicine from the University of Rostock, Rostock, Germany, in 1998 and 2003, respectively. Since 2004, she has been a Collaborator with the Institute of Occupational and Social Medicine, Rostock, Germany. She is currently with the Institute of Preventive Medicine, Faculty of Medicine, University of Rostock.

Kerstin Thurow received the Dip.Chem. degree in chemistry from the University of Rostock, Rostock, Germany, far below the standard period time of study, the Ph.D. degree from the Ludwig Maximilian University, Munich, Germany, under Prof. Dr. Lorenz, working on metal–organic sulfur compounds. In 1999, she finished her habilitation, and was a Faculty Member of the Department of Electrical Engineering, University of Rostock, with a veni legendi for measurement and control. In October 1999, she was appointed as Germany’s youngest University Professor and obtained the worldwide unique professorship for laboratory automation. In December 2004, she was appointed for a novel professorship for life-science automation, where this position is connected with the Center for Life Science Automation (CELISCA) management directorate. Currently, she is the Managing Director of the Institute of Automation, University of Rostock, the Managing Director of CELISCA, and also the President of the Institute for Measurement and Sensor Systems e.V. As a Founding Member and President of the Rostock–Raleigh e.V.—a sister city association—she is striving for cultural, sportive, but also scientific and economical relations to one of the most important life-science regions in the U.S. Dr. Thurow is the recipient of numerous awards, such as for the foundation of the start-up company Amplius—Screening Technologies and Analytical Measurement. In 2004, she was awarded the highly renowned Joachim Jungius Award of Science.

Norbert Stoll received the Dipl.Ing. degree in automation engineering and the Ph.D. degree in measurement technology from the University of Rostock, Rostock, Germany, in 1979 and 1985, respectively. He was the Head of the Department of Analytical Chemistry, Central Institute for Organic Chemistry, Academy of Sciences of German Democratic Republic, Berlin, Germany, until 1991. From 1992 to 1994, he was the Associate Director of the Institute of Organic Catalysis, Rostock. From 1994 to 2000, he directed the Institute of Automation, University of Rostock, where he is currently a Professor of measurement technology with the College of Computer Science and Electrical Engineering. Since 2003, he has also been the Vice President of the Center for Life Science Automation, Rostock. His fields of interests include medical process measurement, lab automation, and smart systems and devices.

Regina Stoll received the Dip.Med. degree, the Dr.Med. in occupational medicine degree, and the Dr.Med.Habil. in occupational and sports medicine degree from the University of Rostock, Rostock, Germany, in 1980, 1984, and 2002, respectively. She is currently a faculty member and the Head of the Institute of Preventive Medicine, Faculty of Medicine, University of Rostock, where she is also a Faculty Member of the College of Computer Science and Electrical Engineering. She is also an Adjunct Faculty Member with the Industrial Engineering Department, North Carolina State University, Raleigh. Her research interests include occupational physiology, preventive medicine, and cardiopulmonary diagnostics.
